- #1

LagrangeEuler

- 708

- 19

[tex]A^{\mu}_{\hspace{0.2cm} \nu}[/tex]

[tex]A^{\hspace{0.2cm} \mu}_{\nu}[/tex]

and

[tex]A^{\mu}_{\nu}[/tex]?

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- Thread starter LagrangeEuler
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- #1

LagrangeEuler

- 708

- 19

[tex]A^{\mu}_{\hspace{0.2cm} \nu}[/tex]

[tex]A^{\hspace{0.2cm} \mu}_{\nu}[/tex]

and

[tex]A^{\mu}_{\nu}[/tex]?

- #2

- 22,035

- 12,945

$${A_{\nu}}^{\mu} = g_{\nu \sigma} g^{\mu \rho} {A^\sigma}_{\rho}.$$

The last expression should be avoided, because the horizontal placement of the indices is not indicated. It's ok if the tensor ##A## is symmetric, i.e., if ##A_{\mu \nu}=A_{\nu \mu}##, because (only!) then

$${A^{\mu}}_{\nu} = g^{\mu \rho} A_{\rho \nu} = g^{\mu \rho} A_{\nu \rho} ={A_{\nu}}^{\mu},$$

and the horizontal ordering is not important.

- #3

martinbn

Science Advisor

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- #4

LagrangeEuler

- 708

- 19

Is there some connection with matrices? For instance, if we have two indices.

$${A_{\nu}}^{\mu} = g_{\nu \sigma} g^{\mu \rho} {A^\sigma}_{\rho}.$$

The last expression should be avoided, because the horizontal placement of the indices is not indicated. It's ok if the tensor ##A## is symmetric, i.e., if ##A_{\mu \nu}=A_{\nu \mu}##, because (only!) then

$${A^{\mu}}_{\nu} = g^{\mu \rho} A_{\rho \nu} = g^{\mu \rho} A_{\nu \rho} ={A_{\nu}}^{\mu},$$

and the horizontal ordering is not important.

[tex]A^{\mu}_{\hspace{0.2cm}\nu}[/tex] what is the row and what is the column? And in this case

[tex]A^{\hspace{0.2cm}\mu}_{\nu}[/tex] what is the row and what is the column?

- #5

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- 10,650

- #6

cianfa72

- 1,246

- 155

Typically when in a given basis you represent a tensor as a matrix the first index (on the left) is the row and the second the column. So in your example ##\nu## is the row and ##\mu## the column.And in this case [tex]A^{\hspace{0.2cm}\mu}_{\nu}[/tex] what is the row and what is the column?

Last edited:

- #7

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